Find the orthogonal trajectories for the family of curves y=(kx)^6.

Answer 1

for our function,

y' = 6k(kx)^5

so, we want other curves whose slope is

y' = -1/(6k^6) x^-5
y = 1/(24k^6 x^4)
or, to make things look more similar,

y = 1/(24k^2) (kx)^-4